Abstract
We consider the problem of finding a curve which interpolates at given points such that (approximately) the length of the curve between each two subsequent interpolation points is equal to some given number. We only consider the functional case. We give an algorithm which yields an interpolating cubic polynomial spline. In case the data is taken from a (smooth enough) function this spline function converges at least quadratically in the mesh size to the original one. If the mesh is ‘regular enough’ it is even third order accurate. We also given an extension to the bivariate case. For the univariate case it will be shown that the length on each interval of this constructed spline at most differs quadratically in the mesh size from the actual lengths. Assuming regularity on the partition this estimate can also be improved by one order.
Zusammenfassung
Wird befassen uns mit dem Problem, eine Kurve zu bestimmen, die in gegebenen Punkten interpoliert und deren Länge zwischen zwei aufeinanderfolgenden Punkten (näherungsweise) gleich einer vorgegebenen Zahl ist. Wir betrachten dabei den Fall, daß die Kurve durch eine Funktion gegeben ist. Es wird ein Algorithmus zur Berechnung eines interpolierenden kubischen Splines vorgestellt. Für Interpolationsdaten von einer (genügend) glatten Funktion ist die Konvergenz der Spline-Funktion mindestens quadratisch in der Maschenweite. Bei (genügend) regulärem Gitter tritt sogar eine Konvergenz der Ordnung drei auf. Wir betrachten ferner den zweidimensionalen Fall. Für den eindimensionalen Fall zeigen wir, daß auch der Fehler des berechneten Splines in der vorgeschriebenen Länge zwischen den Gitterpunkten quadratisch mit der Maschenweite abnimmt. Auch hier erreicht man bei regulärem Gitter eine Erhöhung der Konvergenzordnung um eins.
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van Damme, K., Wang, R.H. Curve interpolation with constrained length. Computing 54, 69–81 (1995). https://doi.org/10.1007/BF02238080
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DOI: https://doi.org/10.1007/BF02238080